It is usually stated in various books or stories 'Energy can't be created or destroyed , but is conserved'. I guess this is within a closed system. Are there any axioms or principles in physics that would be contradicted if certain types of energy could be destroyed or changed into some form that is 'unusable' by any closed or open system? Are the conservation laws axioms? Saying the principles of energy conservation are valid approximations given what is presently known is only a partial answer to this question. I'm wondering if any principles are known now that would determine energy can not be destroyed in many situations not 'covered' by such approximations. Could conservation laws be formulated 'a priori' or without empirical evidence?
2 Answers
Noether's Theorem states that every conservation law must correspond with a symmetry; if one goes through all the math it turns out that energy conservation is related to time symmetry of physical laws (more precisely, the Lagrangian).
So we're relying on the fact that the laws of physics are symmetric with respect to time. Since we've never observed the laws of physics changing with respect to anything at all, we can pretty safely assume that energy is conserved.
Both of these are just empirically observed things, but they're very well-grounded.
I'll refer you to the wikipedia article for details, because it does it better than I can. https://en.wikipedia.org/wiki/Noether%27s_theorem#Example_1:_Conservation_of_energy
(fun fact: if the laws of physics are symmetric with respect to position, too, then momentum is conserved. In X, Y, and Z directions, respectively.)
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This is completely false. Energy can both be destroyed and created. For example, if two gamma rays (photons) interact with a nucleus, they form one electron and one positron, i.e. $\gamma + \gamma \Rightarrow e^- + e^+$. This is also known as pair production. In contrast, if a particle meets its antiparticle they get "converted" into a force carrier particle, such as a gluon, W/Z force carrier particle, or a photon.
The simplest reaction would be if one electron met its antiparticle, a positron. The reaction, as you might guess, ends up in two photons, more specifically in the range of gamma rays. $e^- + e^+ \Rightarrow \gamma + \gamma$. We call this particle annihilation.
But… the total energy and the energy equivalence of the mass in a given system is said as far as we know always constant. In order words, $E + \sum\limits_{k=1}^n \sqrt{{m_k}^2c^4 + \mathbf{p}_k c^2} = \text{constant}$, where $E$ is the total amount of "direct" energy (chemical, electric, radiant, etc.) in the system , $m_k$ the mass of any particle with mass, $c$ the speed of light in vacuum and $\mathbf{p}_k$ the momentum ($\mathbf{p} = \frac{mv}{\sqrt{1-{(\frac{v}{c})}^2}}$) for any particle with mass.
(For $\mathbf{p} = 0$ this long mess above simply becomes the famous $E + \sum\limits_{k=1}^n m_k c^2 = \text{constant}$ from $E = mc^2$)
Of course, we can't be sure of that either. It could be the case that energy can enter and leave from and to other Universes, or that quantum mechanics allows for energy to change within a system.
EDIT: As some already mentioned, quantum mechanics can in fact temporarily violate this principle. In Newtonian mechanics though, the law of conservations is a result of Newton's Laws of Motion. I've found Wikipedia's proof simple and easy to understand:
Suppose, for example, that two particles interact. Because of the third law, the forces between them are equal and opposite. If the particles are numbered 1 and 2, the second law states that $F_1 = \frac{\Delta \mathbf{p}_1}{\Delta t}$ and $F_2 = \frac{\Delta \mathbf{p}_2}{\Delta t}$. Therefore $\frac{\Delta \mathbf{p}_1}{\Delta t} = - \frac{\Delta \mathbf{p}_2}{\Delta t},$ or $\frac{\Delta}{\Delta t} \left(\mathbf{p}_1+ \mathbf{p}_2\right)= 0.$
If the velocities of the particles are $u_1$ and $u_2$ before the interaction, and afterwards they are $v_1$ and $v_2$, then $m_1 u_{1} + m_2 u_{2} = m_1 v_{1} + m_2 v_{2}.$
This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero. This conservation law applies to all interactions, including collisions and separations caused by explosive forces.
Source for quote above: Wikipedia
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